Control moment gyros (CMGs), as applied to spacecraft attitude control and momentum management, have been extensively studied during the past three decades, Refs. 1-8, and more recently in Refs. 9-17. They have been successfully employed for a variety of space missions, such as the Skylab, the MIR station, and the International Space Station (ISS). However, CMGs have never been used in commercial communications and imaging satellites because their higher torque capabilities have not been needed by most commercial and imaging satellites and also because CMGs are much more expensive and mechanically complex than reaction wheels.
A CMG contains a spinning rotor with large, constant angular momentum, but whose angular momentum vector direction can be changed with respect to the spacecraft by gimballing the spinning rotor. Such a CMG is the Astrium/Teldix DMG 15-45S. The spinning rotor is mounted on a gimbal (or a set of gimbals), and torquing the gimbal results in a precessional, gyroscopic reaction torque orthogonal to both the rotor spin and gimbal axes. The CMB is a torque amplification device because small gimbal torque input produces large control torque output on the spacecraft. Because the CMGs are capable of generating large control torques and storing large angular momentum over long periods of time, they have been employed for attitude control and momentum management of large space vehicles, such as the International Space Station (ISS). Four parallel mounted double-gimbal CMGs with a total weight of about 2400 LB and with a design life of 10 years are employed on the ISS. In particular, Kennel's steering law (Ref. 4) has been implemented on the double-gimbaled CMG system of the ISS.
Most of the next-generation commercial imaging satellites may be equipped with CMGs because such satellites will require rapid rotational maneuverability for high-resolution images (Refs. 18-22). In the interest of higher resolution images, a narrower field of vision will be required. To be able to image points of interest more widely distributed than points within the narrow angle of vision of the satellite, rapid slewing movement will be needed. Rather than sweeping the imaging system from side-to-side, the whole spacecraft body will turn rapidly. Pointing the entire spacecraft allows the body-fixed imaging system with a narrow field of view to achieve a higher definition and improves the resolution for its images. The overall cost and effectiveness of such agile spacecraft is greatly affected by the average retargeting time. Thus, the development of a low-cost attitude control system employing smaller and inexpensive CMGs, called mini-CMGs, is of current practical importance for developing future agile imaging spacecraft (Refs. 19-21) as well as small agile satellites (Ref. 22).
The use of CMGs necessitates the development of CMG steering logic which generates the CMG gimbal rate commands in response to the CMG torque commands. One of the principal difficulties in using CMGs for spacecraft attitude control and momentum management is the geometric singularity problem in which no control torque is generated for the commanded control torques along a particular direction. At such a singularity, CMG torque is available in all but one direction. The problem of overcoming singularities in CMG systems has previously been addressed. Approaches to the problem have been largely, but not entirely, successful.
The CMG singularity problem, studied previously by Margulies and Aubrun (Ref. 2) and Bedrossian et al. (Refs. 5 and 6), has been further examined recently in Ref. 23, incorporated herein by reference, to characterize and visualize the physical as well as mathematical nature of the singularities, singular momentum surfaces, and other singularity related problems. The steering logic that may be considered “baseline” steering logic for CMGs is pseudoinverse steering logic, discussed below. It is not free of singularities.
A simple yet effective way of passing through, and also escaping from, any internal singularities, as applied to agile spacecraft pointing control, has been developed in Refs. 13-16, incorporated herein by reference. The CMG steering logic developed and patented by Wie, Bailey and Heiberg of Refs. 13 and 14 is mainly intended for typical reorientation maneuvers in which precision pointing or tracking is not required during reorientation maneuvers, and it fully utilizes the available CMG momentum space in the presence of any singularities. Although there are special missions in which prescribed attitude trajectories are to be “exactly” tracked in the presence of internal singularities, most practical cases will require a tradeoff between robust singularity transit/escape and the resulting, transient pointing errors.
Because the singularity-robust steering logic developed in Refs. 13 and 14 is based on the minimum two-norm, pseudoinverse solution, it does not explicitly avoid singularity encounters. Rather it approaches and rapidly transits unavoidable singularities whenever needed. It effectively generates deterministic dither signals when the system becomes near singular. Any internal singularities can be escaped for any nonzero constant torque commands using the singularity-robust steering logic.
However, the patented CMG steering logic of Refs. 13 and 14 is unable to escape the saturation singularities of certain CMG configurations, which can be problematic if CMG momentum desaturation is desired. Consequently, a new steering logic is desirable to overcome such a deficiency of the singularity-robust CMG steering logic. Furthermore, the new steering logic should provide an effective means of explicitly avoiding, instead of passing through, the internal elliptic singularities that are commonly encountered by most other pseudoinverse-based steering logic.
Control Moment Gyro Systems
There follows a summary of several representative CMG systems. Detailed descriptions of these systems can be found in Ref. 23. These CMG systems will be used in connection with a description of specific exemplary embodiments to demonstrate the simplicity and effectiveness of the new steering logic.
Pyramid Array of Four Single-Gimbal CMGs
For a typical pyramid mount of four single-gimbal CMGs with skew angle of β, shown in FIG. 1, the total CMG momentum vector is represented by                                                         H              =                            ⁢                                                                    h                    1                                    ⁡                                      (                                          x                      1                                        )                                                  +                                                      h                    2                                    ⁡                                      (                                          x                      2                                        )                                                  +                                                      h                    3                                    ⁡                                      (                                          x                      3                                        )                                                  +                                                      h                    4                                    ⁡                                      (                                          x                      4                                        )                                                                                                                          =                            ⁢                                                [                                                                                                                                                                                                                                          -                                  c                                                                ⁢                                                                                                                                   ⁢                                βsin                                ⁢                                                                                                                                   ⁢                                                                  x                                  1                                                                                                                                                                                                                                        cos                                ⁢                                                                                                                                   ⁢                                                                  x                                  1                                                                                                                                                                                                                                                                                  s                          ⁢                                                                                                           ⁢                          β                          ⁢                                                                                                           ⁢                          sin                          ⁢                                                                                                           ⁢                                                      x                            1                                                                                                                                ]                                +                                  [                                                                                                                                                                                                                                          -                                  cos                                                                ⁢                                                                                                                                   ⁢                                                                  x                                  2                                                                                                                                                                                                                                                                          -                                  c                                                                ⁢                                                                                                                                   ⁢                                β                                ⁢                                                                                                                                   ⁢                                sin                                ⁢                                                                                                                                   ⁢                                                                  x                                  2                                                                                                                                                                                                                                                                                  s                          ⁢                                                                                                           ⁢                          β                          ⁢                                                                                                           ⁢                          sin                          ⁢                                                                                                           ⁢                                                      x                            2                                                                                                                                ]                                +                                ⁢                                  [                                                                                                                                                                                                        c                                ⁢                                                                                                                                   ⁢                                β                                ⁢                                                                                                                                   ⁢                                sin                                ⁢                                                                                                                                   ⁢                                                                  x                                  3                                                                                                                                                                                                                                                                          -                                  cos                                                                ⁢                                                                                                                                   ⁢                                                                  x                                  3                                                                                                                                                                                                                                                                                  s                          ⁢                                                                                                           ⁢                          β                          ⁢                                                                                                           ⁢                          sin                          ⁢                                                                                                           ⁢                                                      x                            3                                                                                                                                ]                                +                                  [                                                                                                                                                                                                        cos                                ⁢                                                                                                                                   ⁢                                                                  x                                  4                                                                                                                                                                                                                                        c                                ⁢                                                                                                                                   ⁢                                β                                ⁢                                                                                                                                   ⁢                                sin                                ⁢                                                                                                                                   ⁢                                                                  x                                  4                                                                                                                                                                                                                                                                                  s                          ⁢                                                                                                           ⁢                          β                          ⁢                                                                                                           ⁢                          sin                          ⁢                                                                                                           ⁢                                                      x                            4                                                                                                                                ]                                                                                        [        1        ]            where xi is the ith gimbal angle, cβ≡cos β, and sβ≡sin β. Then we obtain{dot over (H)}=A{dot over (x)}  [2]where {dot over (x)}=(x1, x2, x3, x4) and A is the Jacobian matrix defined as                     A        =                  [                                                                                          -                    c                                    ⁢                                                                           ⁢                  β                  ⁢                                                                           ⁢                  cos                  ⁢                                                                           ⁢                                      x                    1                                                                                                sin                  ⁢                                                                           ⁢                                      x                    2                                                                                                c                  ⁢                                                                           ⁢                  β                  ⁢                                                                           ⁢                  cos                  ⁢                                                                           ⁢                                      x                    3                                                                                                                    -                    sin                                    ⁢                                                                           ⁢                                      x                    4                                                                                                                                            -                    sin                                    ⁢                                                                           ⁢                                      x                    1                                                                                                                    -                    c                                    ⁢                                                                           ⁢                  β                  ⁢                                                                           ⁢                  cos                  ⁢                                                                           ⁢                                      x                    2                                                                                                sin                  ⁢                                                                           ⁢                                      x                    3                                                                                                c                  ⁢                                                                           ⁢                  β                  ⁢                                                                           ⁢                  c                  ⁢                                                                           ⁢                  o                  ⁢                                                                           ⁢                  x                  ⁢                                                                           ⁢                                      χ                    4                                                                                                                        s                  ⁢                                                                           ⁢                  β                  ⁢                                                                           ⁢                  cos                  ⁢                                                                           ⁢                                      x                    1                                                                                                s                  ⁢                                                                           ⁢                  β                  ⁢                                                                           ⁢                  cos                  ⁢                                                                           ⁢                                      x                    2                                                                                                s                  ⁢                                                                           ⁢                  β                  ⁢                                                                           ⁢                  cos                  ⁢                                                                           ⁢                                      x                    3                                                                                                s                  ⁢                                                                           ⁢                  β                  ⁢                                                                           ⁢                  cos                  ⁢                                                                           ⁢                                      x                    4                                                                                ]                                    [        3        ]            Equation (2) represents a linear mapping from {dot over (x)}=(x1, x2, x3, x4) to {dot over (H)}=(Hx, Hy, Hz).
Two different types of the internal singularity for the pyramid array of four CMGs are illustrated in FIGS. 1(a) and (b). The well known, troublesome elliptic singularity at (−90, 0, 90, 0) deg, which cannot be escaped by null motion, is illustrated in FIG. 1(a). The hyperbolic singularity at (−90, 0, 90, 180) deg, which can be escaped by null motion, is shown in FIG. 1(b). Detailed analyses and discussions of CMG singularities can be found in Refs. 5 and 23.
When β=90 deg. FIG. 1 is a 4-CMG configuration with two orthogonal pairs of scissored CMGs. The Jacobian [3] becomes:   A  =      [                            0                                      sin            ⁢                                                   ⁢                          x              2                                                0                                                    -              sin                        ⁢                                                   ⁢                          x              4                                                                                      -              sin                        ⁢                                                   ⁢                          x              1                                                0                                      sin            ⁢                                                   ⁢                          x              3                                                0                                                  cos            ⁢                                                   ⁢            x                                                cos            ⁢                                                   ⁢                          x              2                                                            cos            ⁢                                                   ⁢                          x              3                                                            cos            ⁢                                                   ⁢                          x              4                                            ]  
This special configuration is also of practical importance, as studied extensively in Refs. 1 and 23. Many other CMG configurations are some variants of this basic arrangement of two orthogonal pairs of two parallel CMGs.
Two and Three Parallel Single-Gimbal CMG Configurations
Two and three single-gimbal CMGs with parallel gimbal axes have been investigated in Refs. 1, 2 and 23 for two-axis control of a spacecraft. For a system of two CMGs without redundancy, the momentum vectors, {right arrow over (H)}1 and {right arrow over (H)}2 move in the (x, y) plane normal to the gimbal axis as shown in FIG. 2. For such “scissored” single-gimbal CMGs, the total CMG momentum vector is simply represented as                     H        =                  [                                                                                          cos                    ⁢                                                                                   ⁢                                          x                      1                                                        +                                      cos                    ⁢                                                                                   ⁢                                          x                      2                                                                                                                                                                sin                    ⁢                                                                                   ⁢                                          x                      1                                                        +                                      sin                    ⁢                                                                                   ⁢                                          x                      2                                                                                                    ]                                    [        4        ]            where a constant unit momentum for each CMG is assumed.
Defining a new set of gimbal angles, α and β, as follows:                               α          =                                                    x                1                            +                              x                2                                      2                          ,                  β          =                                                    x                2                            -                              x                1                                      2                                              [        5        ]            where α is called the “rotation” angle and β the “scissor” angle (Ref 1), one obtains                     H        =                  2          ⁡                      [                                                                                cos                    ⁢                                                                                   ⁢                    αcos                    ⁢                                                                                   ⁢                    β                                                                                                                    sin                    ⁢                                                                                   ⁢                    αcos                    ⁢                                                                                   ⁢                    β                                                                        ]                                              [        6        ]            and{dot over (H)}=A{dot over (x)}  [7]where {dot over (H)}=({dot over (H)}x, {dot over (H)}y),{dot over (x)}=({dot over (α)}, β), and A is the Jacobian matrix defined as                     A        =                  2          ⁡                      [                                                                                                                              -                        sin                                            ⁢                                                                                           ⁢                      α                      ⁢                                                                                           ⁢                      cos                      ⁢                                                                                           ⁢                      β                                        ,                                                                                                              -                      cos                                        ⁢                                                                                   ⁢                    α                    ⁢                                                                                   ⁢                    sin                    ⁢                                                                                   ⁢                    β                                                                                                                                          cos                      ⁢                                                                                           ⁢                      α                      ⁢                                                                                           ⁢                      cos                      ⁢                                                                                           ⁢                      β                                        ,                                                                                                              -                      sin                                        ⁢                                                                                   ⁢                    α                    ⁢                                                                                   ⁢                    sin                    ⁢                                                                                   ⁢                    β                                                                        ]                                              [        8        ]            
For a system of three single-gimbal CMGs with parallel gimbal axes, the total CMG angular momentum vector is:   H  =      [                                                      cos              ⁢                                                           ⁢                              x                1                                      +                          cos              ⁢                                                           ⁢                              x                2                                      +                          cos              ⁢                                                           ⁢                              x                3                                                                                                    sin              ⁢                                                           ⁢                              x                1                                      +                          sin              ⁢                                                           ⁢                              x                2                                      +                          sin              ⁢                                                           ⁢                              x                3                                                          ]  and the Jacobian matrix for x=(x1, x2, x3) becomes   A  =      [                                                      -              sin                        ⁢                                                   ⁢                          x              1                                                                          -              sin                        ⁢                                                   ⁢                          x              2                                                                          -              sin                        ⁢                                                   ⁢                          x              3                                                                        cos            ⁢                                                   ⁢                          x              1                                                            cos            ⁢                                                   ⁢                          x              2                                                            cos            ⁢                                                   ⁢                          x              3                                            ]  
The singular momentum surfaces of this systems of three parallel single-gimbal CMGs are described by two circles as shown in FIG. 4. The internal hyperbolic singularity is shown in FIG. 4(a) and the external (saturation) elliptic singularity in FIG. 4(b). Detailed singularity analysis of this system can be found in Refs. 2 and 23.
Four Parallel Double-Gimbal CMGs
For a double-gimbal control moment gyro (DGCMG), the rotor is suspended inside two gimbals and consequently the rotor momentum can be oriented on a sphere along any direction provided no restrictive gimbal stops. For the different purposes of redundancy management and failure accommodation, several different arrangements of DGCMGs have been developed, such as three orthogonally mounted DGCMGs used in the Skylab and four parallel mounted DGCMGs employed for the International Space Station.
As shown by Kennel (Ref. 4), mounting of DGCMGs of unlimited outer gimbal angle freedom with all their outer gimbal axes parallel allows drastic simplification of the CMG steering law development in the redundancy management and failure accommodation and in the mounting hardware.
Such a parallel mounting arrangement of four double-gimbal CMGs with the inner and outer gimbal angles, αi and βi of the ith CMG, is shown diagrammatically in FIG. 3. The total CMG momentum vector H=(Hx, Hy, Hz) is expressed in the (x, y, z) axes as                     H        =                  [                                                                      ∑                                                                           ⁢                                      sin                    ⁢                                                                                   ⁢                                          α                      i                                                                                                                                            ∑                                                                           ⁢                                      cos                    ⁢                                                                                   ⁢                                          α                      i                                        ⁢                    cos                    ⁢                                                                                   ⁢                                          β                      i                                                                                                                                            ∑                                                                           ⁢                                      cos                    ⁢                                                                                   ⁢                                          α                      i                                        ⁢                    sin                    ⁢                                                                                   ⁢                                          β                      i                                                                                                    ]                                    [        9        ]            where a constant unit momentum is assumed for each CMG. The time derivative of H becomes                               H          .                =                  [                                                                      ∑                                                                           ⁢                                      cos                    ⁢                                                                                   ⁢                                          α                      i                                        ⁢                                                                  α                        .                                            i                                                                                                                                            ∑                                      (                                                                                            -                          sin                                                ⁢                                                                                                   ⁢                                                  α                          i                                                ⁢                        cos                        ⁢                                                                                                   ⁢                                                  β                          i                                                ⁢                                                                              α                            .                                                    i                                                                    -                                              cos                        ⁢                                                                                                   ⁢                                                  α                          i                                                ⁢                        sin                        ⁢                                                                                                   ⁢                                                  β                          i                                                ⁢                                                                              β                            .                                                    i                                                                                      )                                                                                                                        ∑                                      (                                                                                            -                          sin                                                ⁢                                                                                                   ⁢                                                  α                          i                                                ⁢                        sin                        ⁢                                                                                                   ⁢                                                  β                          i                                                ⁢                                                                              α                            .                                                    i                                                                    -                                              cos                        ⁢                                                                                                   ⁢                                                  α                          i                                                ⁢                        cos                        ⁢                                                                                                   ⁢                                                  β                          i                                                ⁢                                                                              β                            .                                                    i                                                                                      )                                                                                ]                                    [        10        ]            Note that the x-axis torque component is not a function of the outer gimbal βi, motions. Consequently, in Kennel's CMG steering law implemented on the International Space Station, the inner gimbal rate commands, {dot over (α)}i, are determined first for the commanded x-axis torque, then the outer gimbal rate commands, {dot over (β)}i, for the commanded y- and z-axis torques.
Typical singularities of a system of four parallel double-gimbal CMGs are illustrated in FIGS. 4(a)-(d). The 4H saturation singularity is an elliptic singularity which cannot be escaped by null motion. The 2H and 0H singularities, shown in FIGS. 4(b) and (c) respectively, are hyperbolic singularities which can be escaped by null motion. A nonsingular configuration but with a zero momentum is also shown in FIG. 4(d).
The various CMG systems described in this section are used below to demonstrate the simplicity and effectiveness of the new singularity escape/avoidance logic according to this invention.
Pseudoinverse Steering Logic
Ignoring the effect of spacecraft angular motion, the instantaneous torque vector, {right arrow over (τ)}, generated by CMG gimbal motion {dot over (x)}i can be defined as                               τ          →                =                                            ⅆ                              H                →                                                    ⅆ              t                                =                                    ∑                              i                =                1                            n                        ⁢                                                   ⁢                                                            ⅆ                                                            h                      →                                        i                                                                    ⅆ                                      x                    i                                                              ⁢                                                x                  .                                i                                                                        [        11        ]            or, in matrix form, as                     τ        =                                            ⅆ              H                                      ⅆ              t                                =                                                    ∑                                  i                  =                  1                                n                            ⁢                                                           ⁢                                                                    ⅆ                                          h                      i                                                                            ⅆ                                          x                      i                                                                      ⁢                                  x                  i                                                      =                          A              ⁢                                                           ⁢                              x                .                                                                        [        12        ]            where A is a 3×n Jacobian matrix and {dot over (x)}=({dot over (x)}1, . . . {dot over (x)}n).
For the given control torque command τ, the gimbal rate command {dot over (x)}, often referred to as the pseudoinverse steering logic, is then obtained as{dot over (x)}=A+τ  [13]whereA+=AT(AAT)−1  [14]This pseudoinverse is the minimum two-norm solution of the following constrained minimization problem:                                                                         min                                                                                      x                  .                                                              ⁢                                                                  x                .                                                    2                    ⁢                                           ⁢          subject          ⁢                                           ⁢          to          ⁢                                           ⁢          A          ⁢                                           ⁢                      x            .                          =        τ                            [        15        ]            where ∥{dot over (x)}∥2={dot over (x)}T{dot over (x)}. Most CMG steering laws determine the gimbal rate commands with some variant of the pseudoinverse of the form of the above expression [14].
The pseudoinverse is a special case of the weighted minimum two-norm solution{dot over (x)}=A+τ where A+=Q−1AT[AQ−1AT]−1  [16]of the following constrained minimization problem:                                                                         min                                                                                      x                  .                                                              ⁢                                                x              .                                            ⁢                                                    2                                                                    Q                                              ⁢                                           ⁢          subject          ⁢                                           ⁢          to          ⁢                                           ⁢          A          ⁢                                           ⁢                      x            .                          =        τ                            [        17        ]            where ∥{dot over (x)}∥Q2={dot over (x)}TQ{dot over (x)} and Q=QT>0. Below, the significance of choosing Q≠I, where I is an identify matrix, is demonstrated.
If rank (A)<m for certain sets of gimbal angles, or equivalently rank (AAT)<m, when A is an m×n matrix, the pseudoinverse does not exist, and it is said that the pseudoinverse steering logic encounters singular states. This singular situation occurs when all individual CMG torque output vectors are perpendicular to the commanded torque direction. Equivalently, the singular situation occurs when all individual CMG momentum vectors have extremal projections onto the commanded torque τ.
Since the pseudoinverse, A+=AT(AAT)−1, is the minimum two-norm solution of gimbal rates subject to the constraint A{dot over (x)}=τ, the pseudoinverse steering logic and all other pseudoinverse-based steering logic tend to leave inefficiently positioned CMGs alone causing the gimbal angles to eventually “hang-up” in anti-parallel singular arrangements. That is, they tend to steer the gimbals toward anti-parallel singular states. Despite this deficiency, the psuedoinverse steering logic, or some variant of pseudoinverse, is commonly employed for most CMG systems because of its simplicity for onboard, real-time implementation.
Nakamura, Y. and Hanafusa, H., in Ref. 24, describe a singularity robust steering logic in which:A#=AT[AAT+λI]−1.  [18]The singularity robust inverse steering logic, {dot over (x)}=A#u, does not become singular; i.e., det└AAT+λI┘≠0. This solves the large majority of singularity problems in CMG steering logic. However, {dot over (x)} becomes zero if det(AAT)=0 and if a control torque is commanded along the singular direction. The CMG system with the singularity robust inverse steering logic can then become trapped in the singular state.Singularity Escape/Avoidance Steering Logic
A simple yet effective way of passing through and escaping any internal singularities was developed by Wie, Bailey and Heiberg (Refs. 13-16). Such a singularity-robust CMG steering logic was mainly intended for typical reorientation maneuvers in which precision pointing or tracking is not required during reorientation maneuvers, and it fully utilizes the available CMG momentum space in the presence of any singularities.
The singularity-robust steering logic of Refs. 13-14 has the following form:{dot over (x)}=A#τ  [19]where                               A          #                =                                                            [                                                                            A                      T                                        ⁢                    P                    ⁢                                                                                   ⁢                    A                                    +                                      λ                    ⁢                                                                                   ⁢                    I                                                  ]                                            -                1                                      ⁢                          A              T                        ⁢            P                    =                                                    A                T                            ⁡                              [                                                      A                    ⁢                                                                                   ⁢                                          A                      T                                                        +                                      λ                    ⁢                                                                                   ⁢                    E                                                  ]                                                    -              1                                                          [        20        ]                        and                                                                             P                          -              1                                ≡          E                =                              [                                                            1                                                                      ɛ                    3                                                                                        ɛ                    2                                                                                                                    ɛ                    3                                                                    1                                                  ɛ                                                                                                  ɛ                    2                                                                                        ɛ                    1                                                                                        1                    1                                                                        ]                    >          0.                                    [        21        ]            The positive scalar λ and the off-diagonal elements εi are to be properly selected such that A τ≠0 for any nonzero constant τ.
Note that there exists always a null vector of A# since rank (A#)<3 for any λ and εi when the Jacobian matrix A is singular. Consequently, a simple way of guaranteeing that A#τ≠0 for any nonzero constant τ command is to continuously modulate εi, for example, as follows:εi=ε0 sin(ωt+φi)  [22]where the amplitude ε0, the modulation frequency ω, and the phases φi need to be appropriately selected. The scalar λ may be adjusted as:λ=λ0 exp(−μdet(AAT))  [23]where λ0 and μ are constants to be properly selected.
It is emphasized that the singularity-robust inverse of the form (20) is based on the mixed, two-norm and weighted least-squares minimization although the resulting effect is somewhat similar to that of artificially misaligning the commanded control torque vector from the singular vector directions. Because the singularity robust steering logic is based on the minimum two-norm, pseudoinverse solution, it does not explicitly avoid singularity encounters but it rather approaches and rapidly transits unavoidable singularities whenever needed. The steering logic effectively generates deterministic dither signals when the system becomes near singular. Any internal singularities can be escaped for any nonzero constant torque commands using the singularity robust steering logic.
However, a certain type of external saturation singularity cannot be escaped for any choice of λ and εi, as shown below and therefore a need exists for further improvement in CMG steering logic to escape and/or avoid all singularities, internal and external.